A sequence is geometric when every term is produced by multiplying the previous term by the same fixed number. That fixed number is called the common ratio. If you divide any term by the one before it and always get the same result, the sequence is geometric. That single rule is the entire test.
The Common Ratio
The common ratio is the defining feature of a geometric sequence. To find it, divide any term by the term directly before it. In the sequence 2, 4, 8, 16, dividing each term by its predecessor gives 2 every time: 4÷2 = 2, 8÷4 = 2, 16÷8 = 2. The common ratio is 2.
This works the same way when terms get smaller. In the sequence 100, 20, 4, 4/5, each term divided by the previous one gives 1/5. The common ratio is 1/5, and each term is one-fifth the size of the one before it.
The common ratio can also be negative. Start with 5 and use a common ratio of -2, and you get 5, -10, 20, -40. The terms alternate between positive and negative because multiplying by a negative number flips the sign each time. The pattern still qualifies as geometric because the ratio between consecutive terms never changes.
One important constraint: neither the first term nor the common ratio can be zero. If the ratio were zero, every term after the first would be zero, which isn’t a meaningful pattern. And if the first term is zero, multiplying by any ratio still produces zero forever.
How Geometric Differs From Arithmetic
The easiest way to recognize a geometric sequence is to compare it with the other common type: arithmetic. An arithmetic sequence has a constant difference between consecutive terms. You add the same number each time. A geometric sequence has a constant ratio. You multiply by the same number each time.
Consider 3, 6, 9, 12. The difference between terms is always 3 (addition), so it’s arithmetic. Now consider 3, 6, 12, 24. The ratio between terms is always 2 (multiplication), so it’s geometric. Both sequences start with 3 and 6, but they diverge from there because one grows by adding and the other by multiplying.
The General Formula
Once you know the first term and the common ratio, you can write a formula for any term in the sequence. If the first term is a₁ and the common ratio is r, the sequence looks like this:
a₁, a₁ · r, a₁ · r², a₁ · r³, …
The nth term is a₁ · r⁽ⁿ⁻¹⁾. The exponent is (n – 1), not n, because the first term is the starting point before any multiplication happens. So the second term has been multiplied by r once, the third term twice, and so on.
For example, the sequence -1, 3, -9, 27 has a first term of -1 and a common ratio of -3. The formula for the nth term is -1 · (-3)⁽ⁿ⁻¹⁾, which lets you jump directly to any position. Want the 10th term? Plug in n = 10 and calculate.
You can also define a geometric sequence recursively, meaning each term is defined in relation to the one before it: aₙ = r · aₙ₋₁. This is just a formal way of saying “multiply the previous term by r to get the next one.”
What the Common Ratio Does to the Sequence
The value of the common ratio controls the behavior of the entire sequence. When r is greater than 1, terms grow larger and larger. The sequence 1, 6, 36, 216, 1296 has a common ratio of 6, and the numbers escalate fast. This is the kind of explosive growth people mean when they describe something as “growing geometrically” or “growing exponentially.”
When r is between 0 and 1, terms shrink toward zero. A sequence starting at 18 with a common ratio of 1/3 produces 18, 6, 2, 2/3, 2/9. Each term is a third of the previous one, so the values get progressively smaller without ever quite reaching zero.
When r is negative, the terms alternate in sign. A common ratio of -2 applied to a first term of 5 gives 5, -10, 20, -40. The absolute size of each term still grows (because |-2| is greater than 1), but the sequence bounces between positive and negative values.
When r equals exactly 1, every term is the same: 5, 5, 5, 5. Technically this satisfies the definition (the ratio between consecutive terms is constant), though it’s not a very interesting sequence.
How to Test Whether a Sequence Is Geometric
If someone hands you a sequence and asks whether it’s geometric, the process is straightforward. Divide the second term by the first, the third by the second, and the fourth by the third. If every division gives you the same number, the sequence is geometric and that number is the common ratio.
- 2, 6, 18, 54: 6÷2 = 3, 18÷6 = 3, 54÷18 = 3. Geometric, with r = 3.
- 5, 10, 15, 20: 10÷5 = 2, 15÷10 = 1.5. The ratios differ, so this is not geometric. (It’s arithmetic, with a constant difference of 5.)
- 100, 20, 4, 4/5: 20÷100 = 1/5, 4÷20 = 1/5, (4/5)÷4 = 1/5. Geometric, with r = 1/5.
The check only fails when the ratios aren’t all identical. Even one pair that produces a different ratio means the sequence isn’t geometric.
Real-World Geometric Sequences
Geometric sequences show up anywhere growth or decay happens by a constant percentage. A salary that increases 2% per year is geometric: a $26,000 starting salary becomes $26,520 after one year, $27,050.40 after two, and $27,591.41 after three. Each year’s salary is 1.02 times the previous year’s, so the common ratio is 1.02.
Radioactive decay works the same way in reverse. If a substance loses half its mass every hour, the remaining amounts form a geometric sequence with a common ratio of 1/2. Compound interest, population growth, the spread of a virus in its early stages, and the depreciation of a car’s value all follow geometric patterns. Any time you hear “grows by X percent” or “shrinks by X percent” per period, you’re looking at a geometric sequence.
When Infinite Geometric Sequences Converge
If you keep adding terms of a geometric sequence forever, something interesting happens depending on the common ratio. When the absolute value of r is less than 1, the terms shrink so quickly that their infinite sum approaches a finite number. The sequence 18, 6, 2, 2/3, 2/9 and so on adds up to a specific value rather than growing without bound.
When the absolute value of r is 1 or greater, the terms either stay the same size or grow, so the sum keeps increasing forever. There’s no finite total. This distinction matters in fields like physics and finance, where infinite geometric sums are used to calculate things like the total distance a bouncing ball travels or the present value of a perpetual stream of payments.